To rewrite the expression [tex]\(x^{\frac{9}{7}}\)[/tex] and find the correct option, we need to manipulate the expression using properties of exponents and roots. Here's the detailed breakdown:
The expression [tex]\(x^{\frac{9}{7}}\)[/tex] suggests a fractional exponent. This can be interpreted in several ways using the properties of exponents and roots.
Let's analyze each option:
Option A: [tex]\(x \sqrt[7]{x}\)[/tex]
- This can be rewritten as [tex]\(x \cdot x^{\frac{1}{7}}\)[/tex].
- Simplifying the expression, we get [tex]\(x^{1 + \frac{1}{7}} = x^{\frac{7}{7} + \frac{1}{7}} = x^{\frac{8}{7}}\)[/tex].
Option A is incorrect as [tex]\(x^{\frac{9}{7}} \neq x^{\frac{8}{7}}\)[/tex].
Option B: [tex]\(\left(\frac{1}{\sqrt[3]{x}}\right)^9\)[/tex]
- First, simplify [tex]\(\frac{1}{\sqrt[3]{x}} = x^{-\frac{1}{3}}\)[/tex].
- Raising it to the power of 9 gives [tex]\((x^{-\frac{1}{3}})^9 = x^{-3}\)[/tex].
Option B is incorrect as [tex]\(x^{\frac{9}{7}} \neq x^{-3}\)[/tex].
Option C: [tex]\(x \sqrt[7]{x^2}\)[/tex]
- This can be rewritten as [tex]\(x \cdot x^{\frac{2}{7}}\)[/tex].
- Simplifying the expression, we get [tex]\(x^{1 + \frac{2}{7}} = x^{\frac{7}{7} + \frac{2}{7}} = x^{\frac{9}{7}}\)[/tex].
Option C is correct as [tex]\(x^{\frac{9}{7}} = x^{\frac{9}{7}}\)[/tex].
Option D: [tex]\(\sqrt[9]{x^7}\)[/tex]
- This can be rewritten as [tex]\((x^7)^{\frac{1}{9}} = x^{\frac{7}{9}}\)[/tex].
Option D is incorrect as [tex]\(x^{\frac{9}{7}} \neq x^{\frac{7}{9}}\)[/tex].
Therefore, the correct option is:
C. [tex]\(x \sqrt[7]{x^2}\)[/tex].
So, the final result is that option C [tex]\(x \sqrt[7]{x^2}\)[/tex] correctly rewrites the expression [tex]\(x^{\frac{9}{7}}\)[/tex].
